• Title, Summary, Keyword: law of the iterated logarithm

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CHUNG-TYPE LAW OF THE ITERATED LOGARITHM OF l-VALUED GAUSSIAN PROCESSES

  • Choi, Yong-Kab;Lin, Zhenyan;Wang, Wensheng
    • Journal of the Korean Mathematical Society
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    • v.46 no.2
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    • pp.347-361
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    • 2009
  • In this paper, by estimating small ball probabilities of $l^{\infty}$-valued Gaussian processes, we investigate Chung-type law of the iterated logarithm of $l^{\infty}$-valued Gaussian processes. As an application, the Chung-type law of the iterated logarithm of $l^{\infty}$-valued fractional Brownian motion is established.

THE LAWS OF THE ITERATED LOGARITHM FOR THE TENT MAP

  • Bae, Jongsig;Hwang, Changha;Jun, Doobae
    • Communications of the Korean Mathematical Society
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    • v.32 no.4
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    • pp.1067-1076
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    • 2017
  • This paper considers the asymptotic behaviors of the processes generated by the classical ergodic tent map that is defined on the unit interval. We develop a sequential empirical process and get the uniform version of law of iterated logarithm for the tent map by using the bracketing entropy method.

A SUPPLEMENT TO PRECISE ASYMPTOTICS IN THE LAW OF THE ITERATED LOGARITHM FOR SELF-NORMALIZED SUMS

  • Hwang, Kyo-Shin
    • Journal of the Korean Mathematical Society
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    • v.45 no.6
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    • pp.1601-1611
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    • 2008
  • Let X, $X_1$, $X_2$, ... be i.i.d. random variables with zero means, variance one, and set $S_n\;=\;{\sum}^n_{i=1}\;X_i$, $n\;{\geq}\;1$. Gut and $Sp{\check{a}}taru$ [3] established the precise asymptotics in the law of the iterated logarithm and Li, Nguyen and Rosalsky [7] generalized their result under minimal conditions. If P($|S_n|\;{\geq}\;{\varepsilon}{\sqrt{2n\;{\log}\;{\log}\;n}}$) is replaced by E{$|S_n|/{\sqrt{n}}-{\varepsilon}{\sqrt{2\;{\log}\;{\log}\;n}$}+ in their results, the new one is called the moment version of precise asymptotics in the law of the iterated logarithm. We establish such a result for self-normalized sums, when X belongs to the domain of attraction of the normal law.

STRONG LAWS FOR ARRAYS OF RANDOM VARIABLES

  • Sung, Soo-Hak
    • Bulletin of the Korean Mathematical Society
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    • v.35 no.4
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    • pp.769-775
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    • 1998
  • In this paper, we obtain an analogue of law of the iterated logarithm for an array of independent, but not necessarily idetically distributed, random variables under some moment conditions of the array.

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LIL FOR KERNEL ESTIMATOR OF ERROR DISTRIBUTION IN REGRESSION MODEL

  • Niu, Si-Li
    • Journal of the Korean Mathematical Society
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    • v.44 no.4
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    • pp.835-844
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    • 2007
  • This paper considers the problem of estimating the error distribution function in nonparametric regression models. Sufficient conditions are given under which the kernel estimator of the error distribution function based on nonparametric residuals satisfies the law of iterated logarithm.

A NEW KIND OF THE LAW OF THE ITERATED LOGARITHM FOR PRODUCT OF A CERTAIN PARTIAL SUMS

  • Zang, Qing-Pei
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.5
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    • pp.1041-1046
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    • 2011
  • Let {X, $X_{i};\;i{\geq}1$} be a sequence of independent and identically distributed positive random variables. Denote $S_n= \sum\array\\_{i=1}^nX_i$ and $S\array\\_n^{(k)}=S_n-X_k$ for n ${\geq}$1, $1{\leq}k{\leq}n$. Under the assumption of the finiteness of the second moments, we derive a type of the law of the iterated logarithm for $S\array\\_n^{(k)}$ and the limit point set for its certain normalization.

On the Law of the Iterated Logarithm without Assumptions about the Existence of Moments for the Sums of Sign-Invariant Random Variables (부호불변(符號不變) 확률변수(確率變數)에 합(合)에 대한 반복대수(反復對數)의 법칙(法則))

  • Hong, Dug-Hun
    • Journal of the Korean Data and Information Science Society
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    • v.2
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    • pp.41-44
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    • 1991
  • Petrov (1968) gave two theorems on the law of the iterated logarithm without any assumptions about the existence of moments of independent random variables. In the present paper we show that the same holds true for sign-invariant random variables.

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On Asymptotic Properties of a Maximum Likelihood Estimator of Stochastically Ordered Distribution Function

  • Oh, Myongsik
    • Communications for Statistical Applications and Methods
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    • v.20 no.3
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    • pp.185-191
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    • 2013
  • Kiefer (1961) studied asymptotic behavior of empirical distribution using the law of the iterated logarithm. Robertson and Wright (1974a) discussed whether this type of result would hold for a maximum likelihood estimator of a stochastically ordered distribution function; however, we show that this cannot be achieved. We provide only a partial answer to this problem. The result is applicable to both estimation and testing problems under the restriction of stochastic ordering.